Yes, there is absolutely a connection.This is the Captain John Fred after whom the English Herrings were named.However, this fact was eventually forgotten as the name was carried on through the generations, because they sometimes made the name "John Frederick" which sounded more proper, but was incorrect considering the source in the surname Fred.
The Benjamin Herring who moved to England was the husband of Sarah Jemima Howard, and father of John Fred Herring born 1795, Blackfriars, Surrey, England, "noted painter of animals" who married Anne Harris.
This Benjamin Herring who moved to England, was born in New York, baptised14 July 1771 at New York Dutch Reformed Church, son of John Fred Herring and Lucretia Van Deursen, and that John Fred Herring was born in 1742 in Curacao in the Dutch Antilles (eventually returning to NY) son of Benjamin Herring and his second wife Wendelina Van Gorcum "Heiress to Jagerswoude".
This Benjamin Herring was himself the son of Benjamin Herring and wife Mary Jansen, she when widowed remarrying next Edward Broene, and when again widowed remarrying Francois-Fran Van Couwenhouven, who mentions Captain John Fred and his wife Martha Herring (sister of Benjamin Herring, Jr.) in his will.
I have considerably more on all of this, as I have made the study of this Benjamin Herring line a major focus of my research. You are correct in your assumption that this was the same Herring line.